We use the formalism of tensor network states to investigate the relation between static correlation functions in the ground state of local quantum many-body Hamiltonians and the dispersion relations of the corresponding low-energy excitations. In particular, we show that the matrix product state transfer matrix (MPS-TM)-a central object in the computation of static correlation functionsprovides important information about the location and magnitude of the minima of the low-energy dispersion relation(s), and we present supporting numerical data for one-dimensional lattice and continuum models as well as two-dimensional lattice models on a cylinder. We elaborate on the peculiar structure of the MPS-TM's eigenspectrum and give several arguments for the close relation between the structure of the low-energy spectrum of the system and the form of the static correlation functions. Finally, we discuss how the MPS-TM connects to the exact quantum transfer matrix of the model at zero temperature. We present a renormalization group argument for obtaining finite bond dimension approximations of the MPS, which allows one to reinterpret variational MPS techniques (such as the density matrix renormalization group) as an application of Wilson's numerical renormalization group along the virtual (imaginary time) dimension of the system.. The overall energy scale is represented by a characteristic velocity (e.g., the Lieb-Robinson velocity related to the norm of the Hamiltonian terms, or some spin-wave velocity) in the system.In theory, the full dispersion relation can be reproduced from the ground state if the map between a local Hamiltonian and its corresponding ground state is bijective. For strictly n-local Hamiltonians (i.e., Hamiltonians for which every term acts only on a finite number n of neighboring sites), such a bijective relation is generically obtained. There the n-site reduced-density matrices (RDMs) of ground states represent extreme points in the convex set of all possible n-site RDMs. The Hamiltonian can then be represented as a hyperplane in the space of such RDMs, and the energy will necessarily be minimized for an extreme point in this set. Each of these points uniquely determines an n-local parent Hamiltonian via the tangent space to the boundary at this point, if the boundary is smooth there [4]. This argument is, however, of very limited practical use as it is computationally almost infeasible to characterize this convex set [5]. Also, the uniqueness is only obtained by restricting to a class of n-local Hamiltonians, and there might exist other + n k ( )-local (with ⩾ k 1) or quasi local Hamiltonians for which this is the exact ground state. One of the main goals of this paper is thus to identify which features of all those Hamiltonians can be captured in the ground state and its correlations.We follow a more practical approach based on local information contained within the ground state, which is naturally accessible through a tensor network representation of the same. A central local object arising in te...
The low-temperature dynamics of quantum systems are dominated by the low-energy eigenstates. For two-dimensional systems in particular, exotic phenomena such as topological order and anyon excitations can emerge. While a complete low-energy description of strongly correlated systems is hard to obtain, essential information about the elementary excitations is encoded in the eigenvalue structure of the quantum transfer matrix. Here we study the transfer matrix of topological quantum systems using the tensor network formalism and demonstrate that topological quantum order requires a particular type of ‘symmetry breaking' for the fixed point subspace. We also relate physical anyon excitations to domain-wall excitations at the level of the transfer matrix. This formalism enables us to determine the structure of the topological sectors in two-dimensional gapped phases very efficiently, therefore opening novel avenues for studying fundamental questions related to anyon condensation and confinement.
We present a modification of Matrix Product State time evolution to simulate the propagation of signal fronts on infinite one-dimensional systems. We restrict the calculation to a window moving along with a signal, which by the Lieb-Robinson bound is contained within a light cone. Signal fronts can be studied unperturbed and with high precision for much longer times than on finite systems. Entanglement inside the window is naturally small, greatly lowering computational effort. We investigate the time evolution of the transverse field Ising (TFI) model and of the S = 1/2 XXZ antiferromagnet in their symmetry broken phases after several different local quantum quenches. In both models, we observe distinct magnetisation plateaus at the signal front for very large times, resembling those previously observed for the particle density of tight binding (TB) fermions. We show that the normalised difference to the magnetisation of the ground state exhibits similar scaling behaviour as the density of TB fermions. In the XXZ model there is an additional internal structure of the signal front due to pairing, and wider plateaus with tight binding scaling exponents for the normalised excess magnetisation. We also observe parameter dependent interaction effects between individual plateaus, resulting in a slight spatial compression of the plateau widths. In the TFI model, we additionally find that for an initial Jordan-Wigner domain wall state, the complete time evolution of the normalised excess longitudinal magnetisation agrees exactly with the particle density of TB fermions.
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